What are examples of approaches to dimensionality reduction of feature vectors?
Dimensionality reduction could be achieved by using an Autoencoder Network, which learns a representation (or Encoding) for the input data. While training, the reduction side (Encoder) reduces the data to a lower-dimension and a reconstructing side (Decoder) tries to reconstruct the original input from the intermediate reduced encoding. 

You could assign the encoder layer output ($L_i$) to a desired dimension (lower than that of the input). Once trained, $L_i$ could be used as a alternative representation of your input data in a lower feature-space, and can be used for further computations.